Let E be a compact connected subset of the complex plane that contains more than one point. Show that the family of meromorphic functions on a domain D that omits E (that is, with range in \mathbb{C}^* \backslash E) is a normal family of meromorphic functions.

The hint in the back of the book says to map \mathbb{C}^* \backslash E conformally onto \mathbb{D}, apply thesis version of Montel's theorem. That version states:

Suppose \mathcal{F} is a family of analytic functions on a domain D such that \mathcal{F} is uniformly bounded on each compact subset of D. Then every sequence in \mathcal{F} has a subsequence that converges normally in D, that is, uniformly on each compact subset of D.

I am still confused on how to prove this. I would appreciate some help on this problem. Thanks.